Author: alan

Wave Momentum

“How do waves have momentum?” is a very good question, but like many good questions it seems to attract a lot of over-confident incomplete answers.

It is in fact true that many kinds of travelling waves do transfer momentum to anything that actually absorbs or reflects them, and the momentum transfer is often proportional to the energy density and speed of the wave; but just stating that something is true is not an explanation of why it is true, and if the mere fact of carrying energy explained why waves have momentum then a moving charged battery would have more momentum than an uncharged one.

Indeed, it is perfectly reasonable to not be immediately convinced that waves have any momentum at all in the direction of propagation. For transverse waves the primary motions are perpendicular to the direction of motion and for compression waves the motions forwards and backwards mostly cancel out. And the fact that we can get pushed inwards by a water wave doesn’t tell us anything about net momentum transfer, since anyone who has experienced that inward push has probably also experienced the outward suction of the receding wave; and although waves seem to bring flotsam in to the shore it is not obvious that this is due to the waves themselves rather than the wind that gives rise to them.

When it comes to the often mentioned pressure and momentum of electromagnetic radiation, while we can see the effect of light pressure on the tails of comets, the derivation from Maxwell’s equations is rarely given completely. Many sources (such as this one) explain how the perpendicular electric and magnetic fields lead to a force on any charged particle that is perpendicular to both of them, but don’t give any proof that this is in the forward direction of wave propagation rather than backwards; and even when such a proof is given it is usually shown just as a formal calculation without any physical motivation as to why it is working.

A google search for “wave momentum” is unfortunately overwhelmed by ads and reviews for a popular brand of volleyball shoes, but if we change the order and/or add words like “electromagnetic” or “water” we do get a lot of useful hits. The best I’ve seen so far is

https://as.nyu.edu/content/dam/nyu-as/as/documents/silverdialogues/SilverDialogues_Peskin.pdf

This gives pretty complete arguments for the momentum content of various kinds of waves, (and also includes examples of waves that carry energy but do not have momentum – which shows that your skepticism is not at all unreasonable). But it is at a fairly high mathematical level and so takes a pretty advanced reader to see the physical motivation for why its results are true.

So what I want to do in the rest of this answer is provide a bit of a handwavy argument to give some physical motivation for the momentum content of one particular kind of wave. It is not to be taken too seriously, but just as a hint of what might actually be shown by a proper detailed analysis.

Consider a rope tied to a wall or post at one end, with you holding the other end and moving it up and down to project a wave along the string. If the wave carries momentum then during at least part of the cycle your hand must be applying a forwards push (or at least a reduced amount of tension compared to the starting situation) – and I suspect that, even before thinking about this, it has indeed felt that way when you tried it. That may of course just be a psychological effect rather than anything real, but perhaps we can think of an actual physical reason for it. When your hand is at the extreme top of its motion the rope near the end you are holding is bent up a bit, and as you move it back down the tension in the rope tends to straighten it (even if you just let it go free rather than pulling it down). This pulls up the lower part of the bend, and to counter that pulls down the part nearest your hand. But this swinging down of the end would, if tension were maintained, cause it to project outwards a bit – and so maintaining the original distance from the far end would require a bit less tension (or equivalently a slight push forward relative to the starting level of tension). As I said, this is not a real argument, but it’s the best I can do short of a proper mathematical proof as given in the paper linked to above.

Source: (1000) Alan Cooper’s answer to How do waves have momentum? – Quora

Sound from the Big Bang

A Quora question asks about the possibility of sound from the “Big Bang”:

I’ve been thinking… if we can look back and see the Big Bang and calculate when the universe was created and can see this because of the light traveling through space at the maximum speed that exists. The moment the Big Bang happened the sound had to be extremely loud and the first and only sound ever created which would still be traveling through space like everything else I would think right? So eventually it would make it to our universe and possibly be the catastrophic event that would destroy earth don’t you think?

It may be even more tempting to ridicule the idea of sound in the vacuum of space than the conception of the Big Bang as a localized event that happened someplace far away. But while the latter is a naive misconception, the former may not be so far fetched – at least if we “stretch” our definition of “sound”.

The theory of the “Big Bang” is not that it happened at some particular place, but rather that it happened *everywhere* 14 billion or so years ago; and just as we are now receiving “light” from when it happened 14billion or so light years away, so also whatever stars and beings evolved from what we see at that distance are just now receiving light that started with the part of the bang that happened right here. But although the “bang” was very hot and bright with lots of high energy short wavelengths, what we “see” of it now is very dim and of long wavelength (actually invisible) microwave radiation because the universe has expanded so much between then and now.

The same attenuation and extension of wavelength would apply also to sound waves if they could actually propagate through the vacuum of space, except that the source of what we would “hear” now would be much closer. In fact, back when the universe was denser there may have been variations of density which behaved like sound waves, but the expansion of the universe would have stretched them out by now so that their remnants would not be like audible sound waves (which in any case cannot exist in the vacuum that now fills most of space). Instead they would correspond to density variations on a much bigger astronomical scale. This may indeed be the source of some of the large scale non-uniformity we see in the density and distribution of gas and galaxies on a cosmic scale, but checking to what extent that is the case is a more serious exercise than just speculating that it may be so.

Source: (1000) I’ve been thinking… if we can look back and see the Big Bang and calculate when the universe was created and can see this because of the light traveling through space at the maximum speed that exists. The moment the Big Bang happened the sound had to be extremely loud and the first and only sound ever created which would still be traveling through space like everything else I would think right? So eventually it would make it to our universe and possibly be the catastrophic event that would destroy earth don’t you think? – Relativity IS Easy – Quora

Relativistic Mass

A Quora question asks: What is the equation that states that an object’s observed mass increases with its velocity?

It depends on what you mean by “an object’s observed mass”.

Nowadays the term “mass” is used exclusively for what used to be called the “rest mass” and is a property of the object alone that is independent of the relative velocity of the observer. So no physicist working today would say that “an object’s observed mass increases with its velocity”.

But there was a time in the past when some physicists used the term “mass” (usually, but not always, qualified with the adjective “inertial” or “relativistic”) to identify the multiplier needed to make a relativistically correct equation having the same form as Newton’s third law $#F=ma#$ (albeit only for the special case where the force and acceleration are parallel to the direction of relative motion between object and observer). So the equation you may be thinking of is $#m_{rel}=\frac{m_0}{\sqrt{1-v^2/c^2}}#$, but it is not a statement about what we now mean by “an object’s mass”. (And even adding the adjective “observed” or “apparent” doesn’t change that, as our observation of the “rest” mass is pretty much just as direct as that of the old “inertial” version.)

Despite many strident claims in other answers that it was “incorrect”, the alternative choice of using the word “mass” for $#m_{rel}#$ was in fact perfectly valid if applied correctly. It just wasn’t very useful because the resulting number $#m_{rel}#$ is not a property just of the object itself but depends also on the observer and is different (with a slightly more complicated formula) for accelerations and forces in directions other than that of the relative motion.

Source: (1000) Alan Cooper’s answer to What is the equation that states that an object’s observed mass increases with its velocity? – Quora

Explaining Relativity Without Equations

Can you explain time dilation and space contraction in relativity without using complex mathematical equations?

Yes. Any decent introductory text on relativity does this – but probably just in one or two sentences before going on to derive the actual formula (for which the apparent level of complexity of the resulting equations may depend on the reader’s experience).
The basic idea is that if two identical side-by-side trains are passing by one another and a light signal is sent when the ends from which it is sent are together, then if the trains are in relative motion the signal will reach the far end of one before the other. So if observers on both trains measure the same speed of light then their units of length and/or time must be different. Working out exactly how the coordinates used by each observer are related to those of the other does involve the use of mathematical formulas and equations, but they are well within the scope of high school algebra so whether or not you call them “complex” is a matter of perspective.

Source: (1000) Alan Cooper’s answer to Can you explain time dilation and space contraction in relativity without using complex mathematical equations? – Quora

Time Contraction

Special Relativity tells us that two inertial observers in relative motion each perceive the other to be ageing more slowly – ie each infers that the tick intervals of the moving clock appear to be dilated. But can time contract as well as dilate?

Yes, but with the proviso that the dilation or contraction is just a description of how the progress of one clock appears relative to another and that two observers will not necessarily agree on which events in their lives are simultaneous – and so can only compare average (rather than instantaneous) clock rates using the total time intervals on their clocks between events where they are together.

Two observers who separate and reunite will agree that the total time experienced by the one that felt more forces of acceleration (or of resistance to gravity if spending time near a massive object) will be less than that experienced by the other. This means that from the point of view of the one who was more accelerated (or spent more time at the bottom of a potential well) the clock of the other appears on average to have been speeded up (ie tick intervals appear contracted), while the one who remains unaccelerated interprets this as meaning that that the other’s clock tic intervals were, on average, dilated.

Source: (1000) Alan Cooper’s answer to Can time contract, as opposed to dilation? – Quora

Quantum “Weirdness”

So far as we can tell everything is probably quantum. But it’s just that some quantum things seem more weird to us than others.

Our minds evolved to cope with situations in which the information most relevant to our survival consists of averages over large numbers of microscopic subsystems. For dealing with such averages the “classical” models of reality that we consider natural are good enough for survival (with the advantage of not requiring too much sophistication of the instincts we follow). It is only in specific (mostly artificial) contexts that the quantum nature of reality becomes relevant; so there has been no need for our instincts to take account of that, and so those instincts tend to be based on the classical approximation – and when that approximation fails it feels to us like “weirdness”.

The weirdness stops, not when things are “not quantum”, but just when (due to the averaging business) their quantum behaviour is well approximated by the classical models that correspond to our evolved instincts.

Source: (1000) Alan Cooper’s answer to There is a lot of weirdness in quantum physics at the sub-atomic level, but why does that weirdness stop once things are not quantum? – Quora

Linear or Not?

Why is QM described as a linear theory and GR non-linear?

Because whoever is doing so is confusing completely different aspects of the two theories.

The linearity of QM has to do with the basic framework of the theory, in which the observables are represented by linear operators on a Hilbert space (as opposed to functions on some space of possible configurations); but the equations relating those observables (such as dA/dt=[H,A]) are generally not linear. However the quantum structure does allow us to predict the effects of non-linear evolution of observables by solving a linear (Schrodinger-type) equation for the quantum state.

The non-linearity of GR is about the relationships between observables, and is no more surprising than the non-linearity of Newtonian gravity or of the theory of fluids and elastic media (though it may be more difficult to find interesting situations with useful linear approximations in the case of the GR field equations than for some other classical theories). However this does not rule out the possibility of somehow eventually finding a Quantum version of GR in which observables correspond to linear operators and in which some linear equation satisfied by the quantum states is equivalent to the non-linear evolution of the observables. One reason this might be very difficult though (or even impossible) is due to the fact that, in GR, space and time coordinates are dependent on other observables in a way that may make them unusable as independent variables in some analogue of the Schrodinger equation.

Source: (1000) Alan Cooper’s answer to Why is QM described as a linear theory and GR non-linear? – Quora

Schrödinger’s Equation for Light

A Quora Question Asks: Is it true that there is no Schrodinger equation for light because the Schrodingers’ equation is only for massive particles, and that only Maxwell’s equations can be used for light?

My answer is that it depends on what you mean by the Schrödinger Equation.

The first time many of us see that name is in reference to the position space wave equation for a single massive particle in non-relativistic quantum mechanics (and that is indeed historically appropriate, but is not the only equation that is correctly attributed to Schrödinger). In terms of this usage, the Schrödinger Equation clearly does not apply to light because the behaviour of light is not Galilean-covariant (and, unlike particles, does not even have a useful Galilean-covariant approximation).

But the name “Schrödinger Equation” is also used for the equation that governs the time evolution of any quantum system (which is basically just a statement of the fact that any unitary representation of the one dimensional translation group has a self adjoint generator – which, in the case of time translations, we identify as corresponding to the total energy of the corresponding spacelike time slice). And in that sense, as noted by Mark John Fernee, (with some further elaboration by Diógenes Figueroa) it certainly does apply to the quantized electromagnetic field and so to light.

There is also a Schrödinger Equation in this second sense for the Quantum Field Theory of electrons and positrons; however the relativistic wave equation for a single free electron is not called a Schrödinger Equation but rather is known as the Dirac Equation instead.

For the case of a single relativistic spinless particle, there is no wave equation that is first order in time, and the best we can do is the second order Klein-Gordon Equation; but there is nonetheless a Schrödinger Equation in the second sense for the corresponding scalar quantum field theory.

There is a sense in which Maxwell’s Equations play a similar role to the Dirac and Klein-Gordon equations (and the first interpretation of Schrödinger’s) as the wave equation for a single photon, but the use of Electric and Magnetic fields to extract a probability density is complicated by having to take account of gauge invariance and polarization state.

Source: (1000) Alan Cooper’s answer to Is it true that there is no Schrodinger equation for light because the Schrodingers’ equation is only for massive particles, and that only Maxwell’s equations can be used for light? – Quora

 

Infinite Wavelength of Stationary Particle

A particle does not have a wave function with respect to itself; but for any observer, the uncertainty principle tells us that if a particle could be known to have any exact velocity (and in particular if it is known to be in the observer’s rest frame with a velocity of zero) then its position would be completely unknown – and in the case of zero velocity this would make the wave function constant. (Of course, in practice we never know either the position or the momentum exactly, and this corresponds to the mathematical fact that the constant amplitude wave-function is not normalizable.)

A typical realistic position space wave function is in the form of a wave packet which has an amplitude representing the probability density multiplied by a complex phase factor which oscillates (or more precisely rotates around the unit circle in the complex plane) at a frequency corresponding to the average observed velocity. As the velocity goes to zero, the wavelength of those phase oscillations goes to infinity and the wavefunction just looks like a bump of almost constant phase. But this infinite wavelength does not mean that the wavefunction is constant, and the shape of its amplitude envelope means that its fourier transform includes contributions from frequencies other than that corresponding to the average observed velocity (and so the momentum space wave function is also a bump with width related to that in position space by the uncertainty principle).

Source: (1000) Alan Cooper’s answer to Debroglie says wavelength is inversely proportional to velocity. But velocity is relative, and it is 0 in that particle’s resting frame of reference. So what does it mean for a particle to have infinite wavelength? – Quora

On Time and Space

A Quora question asks: Why are conservation laws related to conservation while systems evolve in time, while there are no equivalent principia that applies to conservation in space? Does this fact constitute an ontologic distinction between time and space?

The distinction between space and time (regardless of whether or not you fancy it up with the silly word “ontologic”) is that from our point of view time is the direction in event-space for which we have memory of one half-space but not the other, and our experience of more recent compared to more distant memories defines for us what it means for systems to evolve. So time for any observer is by definition the direction in which systems evolve for that observer (and this is true regardless of what laws, conservation or otherwise, we use to predict that evolution).

Before the 20th century it was generally assumed that the time direction was the same for all observers, but we now know that it depends to some extent on the observer. Fortunately the evolution of physical systems can be summarized in laws which appear to have the same form for all observers, but the fact that some of these are conservation laws is quite irrelevant to the distinction between time and space.

Some systems do have the property that their structure in one spatial direction can be inferred from knowledge of that in the opposite direction, and in some cases there are quantities that are the same for all values of the relevant coordinate. We don’t normally refer to changes with respect to a spacelike parameter as “evolution”, and although quantities that are independent of position might well be said to be “conserved over space” we generally understand the unmodified word “conserved” to refer to conservation over time.

Source: (1000) Alan Cooper’s answer to Why are conservation laws related to conservation while systems evolve in time, while there are no equivalent principia that applies to conservation in space? Does this fact constitute an ontologic distinction between time and space? – Quora